Rohit Kumar

Rohit Kumar

RMIT University

Biography

My name is Rohit and I am currently an Honours student in Mathematics at the School of Mathematical and Geospatial Sciences, RMIT University.

I enrolled as a Bachelors of Science (Mathematics) student at RMIT in semester 2, 2011 and completed this degree in July, 2014, majoring in Information Security. As an undergraduate student, I completed a group project on Rainfall Prediction and two placements: Mathematics and Statistics in Industry Group (MISG) in February, 2012 and in DSTO (Defence Science and Technology Organisation), Edinburgh, South Australia in 2013. My placement in DSTO involved adapting a detection scheme to Pareto Distribution in order to handle multiple interfering targets and clutter transitions between aircraft. Pareto Distribution is a radar intensity distribution that is used in High Resolution Radar to model clutters. Part of fulfilling this task involved deriving detection probability equations and simulating them using MATLAB. This placement helped me gain an insight into the Radar field and motivated me to perhaps pursue research as a career.

I am also a member of the Golden Key international International Honours Society and APESMA (Association of Professional Engineers, Scientists and Managers Australia).

My interests in Applied Mathematics include Linear Algebra, Differential Equations, Computational Mathematics, Mathematical Modelling and Real and Complex Analysis. I have also developed an interest in Applied Cryptography such as cyber security, mobile security, Viterbi decoding and so on. Applied Cryptography is an area I may consider to pursue research in.

Apart from my academic interests, I have various other outlets such as playing cricket, listening and practicing music, watching sports and reading statistics in sports. I perform in South Indian Classical Music concerts on the Indian Drums and play the Western Drums.

Smith’s Population Model in a Slowly Varying Environment

The standard single species logistic population model proposed by Verhulst and later rediscovered by Pearl and Reed has long been a successful workhorse for representing the evolution of populations. In 1963, F E Smith, in his study of water fleas, reported that the logistic model was not capable of fitting the experimental data, due to time lags in the effects of population density on birth and death rates. This caused distortion in the resulting population density evolution curves. He proposed a modified model which took time lags into account through the introduction of an extra positive parameter C. Unlike the logistic model, the differential equation of this model cannot be solved explicitly, but a solution may be obtained in implicit form.An added complication occurs in both models when the parameters defining the model are no longer constants, but vary with time. This would be so when they are considered in a varying environment. While solutions may exist when the parameters vary in specific ways, for completely arbitrary time variation, exact solution of either of the DEs in both models is rarely possible, and we must resort to numerical solutions. These have the disadvantage of requiring explicit parameter values and are of limited use in studying general trends. However, when the parameters are slowly varying functions of time, multitiming techniques may be used to obtain useful approximations for the evolving population and lead to a deeper understanding of the solution structure.This project will investigate Smith’s model when all parameters vary slowly with time in a prescribed manner. The multitiming techniques referred to above will be applied to yield approximate expressions for the evolving population. In particular these will be developed for a range of slowly varying C, and will be compared with those generated numerically, using a solver such as Maple. Extension to the case where all model parameters vary slowly will be considered if time permits.

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